# Supernode Analysis

In nodal analysis, when a voltage source comes in between two nodes not involving the reference node then supernode analysis has to be performed. Nodal analysis is a method of network analysis particularly suited for the networks having many parallel circuits with a common ground connected node. This method of network analysis is based on Kirchhoff’s first law or KCL.

Nodal analysis is also termed as node-voltage theorem. Nodal analysis significantly reduces the computational work by reducing the set of equations which we need to solve in a particular circuit or network.

## Illustration of Supernode Analysis

Let us consider an electrical circuit as shown below.

The circuit above consists of four node. The node ‘O’ is taken as reference node and the other three nodes are labeled with voltages V_{1}, V_{2} and V_{3}. Here, a voltage source V_{S} is present between the two nodes 1 and 3. As this voltage source V_{S} is present between two nodes and also it does not involves any reference node, now we need the concept of the supernode.

For the supernode analysis as voltage source comes between node 1 and 3. So node 1 and 3 are taken as supernode.

For the analysis of above circuit using the nodal analysis, we have three unknown node voltage V_{1}, V_{2} and V_{3}. So, we have to form three linear equation in order to compute the unknown voltages.

### Step 1:

__First formulate the equation for the node which does not falls under supernode.__ Such as in above case node 2 does not fall under supernode and we can easily write KCL (Kirchhoff’s Current Law) for this node.

The KCL equation for the node 2 is given by

### Step 2:

__Secondly, formulate voltage equation from the voltage source present between two nodes. __Such as in above case the equation can be formed from the voltage source V_{S} present between the node 1 and 3.

The equation will be given by

### Step 3:

__Thirdly, write the KCL equation for the supernode. While writing the KCL equation for supernode do not consider the branch of the circuit which contains the voltage source.__ Such as in the above circuit, write KCL equation for supernode 1 and 3 and while writing the equation do not consider the branch containing the voltage source V_{S}.

The KCL equation for the supernode will be

__Finally the system of linear equations can be solved to find the unknown node voltages.__

## Example

Here in the given circuit a voltage source of 36 Volts is present between two nodes 1 and 2 so we have to perform supernode analysis.

The voltage of the node 1 and 2 are V_{1} and V_{2} respectively. The reference node is highlighted with yellow color.

First the equation is formed from the voltage source present between two nodes.

Secondly, we will write the KCL equation for the supernode 1 and 2.

Solving the equation (i) and (ii) we have,

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